As a worst case situation, we may consider that the FC has no a-priori information about the distribution of Byzantines. This corresponds to maximizing the entropy of \(A^n\) , i.e. to assuming that all sequences \(a^n\)  are equiprobable, \(P(a^n) = 1/2^n\) . In this case, the random variables \(A_i\)  are independent of each other and we have \(P_{A_i}(0) = P_{A_i}(1) = 1/2\) . It is easy to argue that in this case the Byzantines may impede any meaningful decision at the FC. To see why, let us assume that the Byzantines decide to use \(P_{mal}=1\) . With this choice, the mutual information between the vector state \(S^m\)  and \({\bf R}\)  is zero and so any decision made by the FC center would be equivalent to guessing the state of the system by flipping a coin. The above observation is consistent with previous works in which it is usually assumed that the probability that a node is Byzantine or the overall fraction of Byzantines is lower than 0.5, since otherwise the Byzantines would always succeed to blind the FC [1]}.
